What is quantum entanglement? Part 4: relativity and entanglement

This is part 4 in a lengthy series of posts attempting to explain the idea of quantum entanglement to a non-physics audience.  Part 1 can be read here,  Part 2 can be read here, and Part 3 here

In the last post, we finally introduced the concept of quantum entanglement.  An example of an entangled state between two quantum particles is given by the decay of a spin-zero pion into a spin-1/2 positron and a spin-1/2 electron, as illustrated below.

This results in a combined quantum spin state for the electron and positron that may be written as:

We may read this as “the two spin-1/2 particles end up in a quantum state which is an equal superposition of the positron being spin-up and the electron being spin-down with the positron being spin-down and the electron being spin-up.”

This suggests that the electron and positron, when produced in the decay, might be considered to exist simultaneously in a state where the electron (-) is up and the positron (+) is down, and vice-versa — their fates are “entangled.”  When we measure the state of one of the particles, say the electron, it is 50% likely to “choose” the spin-up state and 50% likely to choose the spin-down state.  When it does, the positron, no matter how far away, must instantly take on the opposite spin state — at least according to the original Copenhagen interpretation of quantum physics. In short, after measurement, the combined state of the electron and positron is either:

But, note the use of the word “instantly.” Because angular momentum is conserved, if the electron is measured spin-down, the positron must be in a spin-up state.  This collapse of the wavefunction must happen as soon as the electron is measured, otherwise there would be the possibility of measuring the positron also in a spin-down state, which would violate angular momentum conservation.

This would seem to suggest that the electron must send a “message” to the positron, and this message arrives instantaneously, regardless of the distance between them.  However, according to Einstein’s special theory of relativity, nothing is supposed to be able to move faster than the vacuum speed of light.

This raises the question: does entanglement violate special relativity?  And, if it does, can we use it to communicate over vast distances at superluminal speeds?

As it turns out, the correct answer is “neither.”  Entanglement, when considered carefully in the context of the full quantum theory, turns out to be perfectly consistent with relativity.  But, as we will see, it is quite a theoretical adventure to come to that conclusion!

We begin our adventure by pointing out that it is well-known that there are, in fact, all sorts of “things” that can move faster than the vacuum speed of light — for a sufficiently loose definition of “things.”  A couple of classic examples will serve to illustrate the point.  For the first example, imagine a tiny gnat flying in front of a laser beam that is being projected at the Moon.  The gnat casts a shadow and, because the beam itself gets wider as it travels, so will the shadow.  This means that the slow-moving gnat on the Earth creates a very fast-moving shadow on the Moon!

A beam of light keeps spreading the further it gets away from the source, which means that the motion of a shadow gets faster the further away it gets. In a previous post, I estimated that, on Pluto, the shadow would be moving at roughly 10 times the speed of light!

Another example is known as the “scissors paradox” in special relativity. You may not have noticed this before, but the intersection point between the scissors blades moves faster as the blades close. This increase in speed is why scissors make a “snip” sound when you close them — they make a low frequency “sss” when the intersection is moving slow, and a high frequency “nip” when it’s moving fast!

If we make a really, really long pair of scissors — say, on the order of several light years in length — we can easily make the intersection point move faster than the speed of light.

Now we ask: can we use these systems to send messages at a speed faster than light?

It is pretty clear that, in the shadow example, the answer is “no.” Imagine that we have two bases on Pluto, and want to use the shadow to send a message from Base A to Base B. In principle, it would seem that we could use the shadow as a message: Base B will definitely know when it is cloaked in shadow. However, because the shadow is controlled by the gnat on the Earth, Base A must first send a message to Earth to tell the gnat to move. This message must travel to Earth at the speed of light, and then the shadow itself will not appear on Pluto until it has propagated from Earth, also at the speed of light. Because the travel time for light from Earth to Pluto is on the order of 5 hours, it will take much, much longer to send a message with a shadow than just by using an ordinary beam of light.  The shadow cannot be used to convey a message without the help of the intermediate “gnat signal,” which travels at the speed of light.

Using shadows to communicate on Pluto. Literally nothing in this figure is to scale.

The scissors example is a little more subtle to explain, but the result is the same: it is still not possible to use them to communicate faster than light.  Suppose a messenger A tries to use our giant scissors to cut a rope at a distant station B, making a bell ring.  The messenger A starts to close the scissors, but the blades themselves will not move instantaneously: the force applied to the blades will travel along them at the speed of light, until the entire pair of scissors is in motion.  Once the blades start moving, the intersection point will move faster than the speed of light, but this will always be lagging behind the motion of the blade tips, which only started moving once the force reached them.

For both our “superluminal” examples, we find that the actual transfer of a message using faster-than-light techniques still always lags behind signals that are sent at the speed of light.

So we need to refine our original statement that “nothing travels faster than the vacuum speed of light.”  Instead, we should say that “nothing useful travels faster than the vacuum speed of light.”  Any sort of trick that we manage to use to make something — shadows, blade crossings — move faster than light seems to end up not being able to carry useful information without first having something moving at the relativistic limit.

We now turn back to our original question: “does entanglement allow faster-than-light communication?”  We may now restate this as: “can we transmit any useful information using entanglement?”

We consider a possible communication scenario of the simplest form to begin with.  Let us imagine a single entangled electron-positron pair is sent to distant observers A and B, each of whom store their particle, unmeasured, until ready to “communicate.”  This is sketched below.

A simple entanglement scheme. The orange wavy line is there to remind you that the particles are connected in a “spooky” quantum way.

When ready, observer A will open the box, and use the analyzer to determine whether her particle is spin-up or spin-down.  This will cause the wavefunction to instantaneously collapse, and put the particle of observer B in the opposite state of that measured by A.

One of two possible outcomes of measurement by observer A. She finds her particle is spin-down, which means that observer B’s particle is now definitely spin-up, even though he hasn’t even looked at it yet.

The first question we ask: can observer B tell from his measurement whether observer A actually performed a measurement on her particle? This would be a way to transmit a simple “yes”  or “no” message: “yes (I’ve measured my particle)” or “no (I haven’t measured my particle).”

The answer (to our physics question, not the measurement) is “no.” When observer A opens her box and uses her analyzer, she sees a particle that is either definitely spin-up or definitely spin-down.  The same is true for observer B.  Regardless of who makes their measurement first, A or B, they each have a 50/50 chance of finding the particle spin-up or spin-down.  The possible outcomes for observer B are the same regardless of whether observer A performs her measurement.  And observer A has no control over the outcome of her measurement, so she has no control over the outcome of observer B’s measurement.  The randomness inherent to quantum mechanics effectively ruins any hope of communication in this simple scheme.

But there is one limitation of our first naive attempt: observer A and observer B both attempt the same type of measurement of their particles.  To be more specific, both of them measure the spin of their respective particles along the vertical axis, using their analyzers oriented vertically.  If observer A measures her particle’s spin along a different axis, this will result in particle B ending up in a different quantum state. Perhaps this difference can be detected, and therefore used to send information?  Observer A could send “yes” or “no” by measuring “vertical” or “horizontal,” if this scheme works.

To explain how this would work, we need to talk a little more about quantum spin and the idea of a change of basis.  We have previously described quantum spin as the inherent angular momentum of a quantum particle, roughly analogous to the Earth spinning around its axis. However, when measured, the spin of an electron can only take on two possible values, +1/2 or -1/2, which we call spin-up and spin-down.  Strangely, we have the same possibilities regardless of how we choose to measure the particle: if we measure the spin vertically, horizontally, or longitudinally, we will always get one of two results: +1/2 or -1/2. This is very different from the measurement of a classical spinning object.  If we were to introduce a coordinate system to describe the rotation of a spinning globe, for instance, we would generally need 3 numbers to describe its rotation, corresponding to how much it is rotating around the x, y and z-axes.  In the case of a spin-1/2 particle, once we’ve measured the spin along any axis, we know exactly what the particle’s spin quantum state is: in fact, thanks to wavefunction collapse, we have transformed the particle into that quantum state.

One implication of the simple nature of quantum spin is that the same quantum states that we use to describe a particle being spin-up or spin-down in a vertical measurement can also be used to describe a particle being spin-left or spin-right in a horizontal measurement.  We can write the actual quantum state of the spin in terms of the up/down states or the left/right states.

In words, we may say: a particle that is in a definite spin-right state or spin-left state may be considered an equal mixture of spin-up and spin-down (with the sign of the sum determining which).  This can be reversed, as well, to say that:

The choice of how we choose to describe the spin state — up/down or left/right — is called the basis.  If we decide to write one set of states in terms of another — left/right in terms of up/down, for example — it is known as a change of basis.

Why does spin work this way, so that we can describe up/down spin in terms of left/right, and vice-versa?  We would need to do some significant math to prove these relations, but they make a bit of intuitive sense.  A particle that is definitely in a horizontal spin state must have equal mixtures of spin-up and spin-down that cancel out, which is exactly what we have above.

Now we come to the key question: what happens if observer A measures her particle with a horizontal analyzer?  That is, our experiment pre-measurement now looks like this:

Let us suppose that observer A measures that the electron is spin-right. That means that the positron is in a definite spin-left state. But observer B is using a vertical analyzer, so they are measuring whether the particle is spin-up or spin-down. That means that the state of the positron sitting in observer B’s box is:

We can roughly illustrate what happens, prior to B’s measurement, by the following image.

The positron is definitely spin-left, but that is equivalent to being in a superposition spin-up and spin-down state.

However, this spin-left state is a state that, again, has a 50% chance of being measured in the spin-up position and 50% chance of being measured in the spin-down position, which is exactly the outcome we would expect if both analyzers were vertical!  Apparently changing the type of measurement that observer A makes will also not effect the measured outcomes that observer B will make.

But wait — there’s one more possibility!  Instead of looking at individual particles, let us imagine that we insert a “quantum duplicator” at the output of observer B’s box, that will create a large number of positrons all in the exact same quantum state as the original one.  In short: when observer B decides to make a measurement, he opens his box, and the positron passes into the quantum duplicator, which makes a large number of exact copies of it.  Then each of those copies goes through the analyzer.

We illustrate all four possible outcomes now — two possible measurements that observer A can make, and two possible outcomes for each of those measurements by observer A.

I’ve boxed the two distinct outcomes above.  When observer A measures the electron with a horizontal analyzer, observer B will get a bunch of positrons in superposition states. When measured, the positrons will be 50% up, 50% down.  However, when observer A measures the electron with a vertical analyzer, observer B will either get positrons 100% up, or 100% down!

This seems to suggest that observer A can signal, superluminally, to observer B by the choice of analyzer orientation.  For instance, “horizontal” could mean “yes” and “vertical” could mean “no.”  This scheme, however, depends on the ability to make a quantum multiplier, that can create multiple perfect copies of a given quantum particle.  Is this possible?

In fact, it is not!  There is a theorem in quantum physics known as the no-cloning theorem that explicitly states that it is not possible to copy the quantum state of a particle without wrecking the original quantum state.  That is, we could make particle #2 have the same state as particle #1 originally had, but in the process we would have changed the state of particle #1.  Making multiple perfect copies of a single quantum state is therefore not possible, and our superluminal communication scheme¹ using entangled photons simply won’t work.

Proving the no-cloning theorem is definitely outside the scope of this post and our abilities, but a little thought about measurement and wavefunction collapse shows why it is plausible.  Wavefunction collapse is an indication that, whenever we make a measurement of a particle’s quantum state, we in general change that quantum state.  Any attempt to “clone” a particle would, in essence, require that the quantum state of that particle be measured², changing it.

In the end, then, we conclude that entanglement does not allow us to communicate faster than light.  Surprisingly, the inability to do so hinges upon the seemingly-unrelated inability to duplicate quantum states! The collapse of the entangled wavefunction falls into the category of “stuff that moves faster than light but isn’t useful” like shadows and giant scissors.

To conclude, I should note again that we have based our entire discussion of superluminal communication on the Copenhagen interpretation of quantum physics, which assumes that the  wavefunction collapses when it is measured.  There has not yet been any experiment that disproves the Copenhagen interpretation, but there are other interpretations of quantum physics that reach the same conclusions. We will discuss some of these interpretations in a future post.

However, in my next entanglement post, I want to talk about how physicists generate entangled quantum states in the laboratory!  There are much, much better and more efficient ways to do it other than with decaying pions.

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¹ This argument was made in a classic paper by D. Diecks, “Communication by EPR devices,” Phys. Lett. 92A (1982), 271-272.  Diecks shows that, if one assumes that a quantum multiplier exists, one runs into a contradiction with fundamental aspects of quantum physics, which implies that the multiplier cannot exist. Note that this paper appeared not too long ago — 1982!  The arguments about entanglement, what it means, and what can be done with it are still ongoing.

² This isn’t really a great way of justifying the no-cloning theorem, because one could imagine the possibility of cloning a particle through quantum interactions without doing a classical measurement.  That is, we could bring 2 or more quantum particles together and hypothetically make them interact in such a way that they come out in the same state.  The no-cloning theorem states that such a cloning process is not possible in any way.  Diecks’ paper, mentioned above, is one of the first mentions of “no-cloning.”

 

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4 Responses to What is quantum entanglement? Part 4: relativity and entanglement

  1. adithya says:

    Great article. Thoroughly enjoyed all four parts. Thank you.

  2. MZH says:

    Quick technical question: what operation is meant by |A>|B>? That is, in contrast to the inner product and the projection operator |A><A|.

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